Maximum of a partial sample in the uniform AR(1) processes

The limit distribution of the random vector , whose components are maximum of the sub-sample with even indices and maximum of the full sample from the uniform AR(1) process with parameter r[greater-or-equal, slanted]2, is determined. For some values un<vn, events and {Mn[less-than-or-equals, slant]vn} are asymptotically independent as n-->[infinity], while for other values un<vn these events can be asymptotically perfectly dependent.